Identifying the stored energy of a hyperelastic structure by using an attenuated Landweber method

Abstract

We consider the nonlinear inverse problem of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field as well as from surface sensor measurements. The displacement field is represented as a solution of Cauchy’s equation of motion, which is a nonlinear elastic wave equation. Hyperelasticity means that the first Piola-Kirchhoff stress tensor is given as the gradient of the stored energy function. We assume that a dictionary of suitable functions is available. The aim is to recover the stored energy with respect to this dictionary. The considered inverse problem is of vital interest for the development of structural health monitoring systems which are constructed to detect defects in elastic materials from boundary measurements of the displacement field, since the stored energy encodes the mechanical properties of the underlying structure. In this article we develop a numerical solver using the attenuated Landweber method. We show that the parameter-to-solution map satisfies the local tangential cone condition. This result can be used to prove local convergence of the attenuated Landweber method in the case that the full displacement field is measured. In our numerical experiments we demonstrate how to construct an appropriate dictionary and show that our method is well suited to localize damages in various situations.